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Stream: event: Online CT seminar

Topic: Tue May 22 1400 GMT -- Claudio PIsani -- Unbiased symmetr...

Eric M Downes (May 16 2024 at 14:06):

Next week @Claudio Pisani will be giving a talk! The talk discussion can go in this topic. The talk will be recorded (links posted in the spreadsheet and the recordings topic) and we encourage everyone to come so you can ask questions. Claudio might also post his abstract here.

Eric M Downes (May 16 2024 at 14:51):

The title of the talk is "Unbiased Symmetric Multicategories" in case zulip is cutting that short for anyone else.

Claudio Pisani (May 16 2024 at 21:01):

Here is the abstract:

We present an unbiased theory of symmetric multicategories, where sequences are replaced by families.
To be effective, this approach requires an explicit consideration of indexing and reindexing of objects and arrows,
handled by the double category $\mathbb{P}\rm{b}$ of pullback squares in finite sets.
A symmetric multicategory is then a finite sum preserving discrete fibration of double categories
$M: \mathbb{M}\to\mathbb{P}\rm{b}$.

The idea is that

• $\mathbb{M}_0$ is the category of families of objects of $M$ with their reindexing;
• $\mathbb{M}_1$ is the category of families of arrows of $M$ with their reindexing;
• loose composition corresponds to composition in $M$.

If the loose part of $M$ is an opfibration (with opcartesian arrows stable with respect to reindexing) we get unbiased symmetric monoidal categories.

As a first immediate generalization, we can remove the finiteness condition to obtain infinitary symmetric multicategories.

More interestingly, we can replace the indexing base $\mathbb{P}\rm{b}$ with another "double prop" $\mathbb{P}$, obtaining the category of $\mathbb{P}$-multicategories as the slice $\cal DP/\mathbb{P}$ of the category of double props.
For instance, we can enhance $\mathbb{P}\rm{b}$ by totally ordering each fiber of its loose arrows to obtain the prop $\mathbb{T}\rm{ot}$ for plain multicategories.
The morphism of props $\mathbb{T}\rm{ot} \to \mathbb{P}\rm{b}$, which is itself a symmetric multicategory, induces then a functor $L: \bf Mlt \to \bf sMlt$ between the corresponding slices and the adjunction $L\dashv R: \bf sMlt \to \bf Mlt$ arises as an instance (when $\cal B = \cal DP$) of the general adjunction between the fibers in the codomain bifibration on a category with pullbacks $\cal B$.

We present other instances and show how several concepts and properties find a natural setting in this framework.

We also consider cartesian multicategories as the algebras for a monad $(-)^{\rm cart}$ on $\bf sMlt$,
where the loose arrows of $\mathbb{M}^{\rm cart}$ are "spans" formed by a tight and a loose arrow in $\mathbb{M}$.